Mass, Density, and Volume - Real-life applications

Photo by: Oleg Shelomentsev

Atomic Mass Units

Chemists do not always deal in large units of mass, such as the mass
of a human body—which, of course, is measured in kilograms.
Instead, the chemist's work is often concerned with
measurements of mass for the smallest types of matter: molecules,
atoms, and other elementary particles. To measure these even in terms
of grams (0.001 kg) is absurd: a single atom of carbon, for instance,
has a mass of 1.99 · 10
−23
g. In other words, a gram is about 50,000,000,000,000,000,000,000
times larger than a carbon atom—hardly a usable comparison.

Instead, chemists use an atom mass unit (abbreviated amu), which is
equal to 1.66 · 10
−24
g. Even so, is hard to imagine determining the mass of single atoms
on a regular basis, so chemists make use of figures for the average
atomic mass of a particular element. The average atomic mass of
carbon, for instance, is 12.01 amu. As is the case with any average,
this means that some atoms—different isotopes of
carbon—may weigh more or less, but the figure of 12.01

A
LTHOUGH
S
ATURN IS MUCH LARGER THAN
E
ARTH
,
IT IS MUCH LESS DENSE
.

amu is still reliable. Some other average atomic mass figures for
different elements are as follows:

Hydrogen (H): 1.008 amu

Helium (He): 4.003 amu

Lithium (Li): 6.941 amu

Nitrogen (N): 14.01 amu

Oxygen (O): 16.00

Aluminum (Al): 26.98

Chlorine (Cl): 35.46 amu

Gold (Au): 197.0 amu

Hassium (Hs): [265 amu]

The figure for hassium, with an atomic number of 108, is given in
brackets because this number is the mass for the longest-lived
isotope. The average value of mass for the molecules in a given
compound can also be rendered in terms of atomic mass units: water (H
2
O) molecules, for instance, have an average mass of 18.0153 amu.
Molecules of magnesium oxide (MgO), which can be extracted from sea
water and used in making ceramics, have an average mass much higher
than for water: 40.304 amu.

These values are obtained simply by adding those of the atoms included
in the molecule: since water has two hydrogen atoms and one oxygen,
the average molecular mass is obtained by multiplying the average
atomic mass of hydrogen by two, and adding it to the average atomic
mass of oxygen. In the case of magnesium oxide, the oxygen is bonded
to just one other atom—but magnesium, with an average atomic
mass of 24.304, weighs much more than hydrogen.

Molar Mass

It is often important for a chemist to know exactly how many atoms are
in a given sample, particularly in the case of a chemical reaction
between two or more samples. Obviously, it is impossible to count
atoms or other elementary particles, but there is a way to determine
whether two items—regardless of the elements or compounds
involved—have the same number of elementary particles. This
method makes use of the figures for average atomic mass that have been
established for each element.

If the average atomic mass of the substance is 5 amu, then there
should be a very large number of atoms (if it is an element) or
molecules (if it is a compound) of that substance having a total mass
of 5 grams (g). Similarly, if the average atomic mass of the substance
is 7.5 amu, then there should be a very large number of atoms or
molecules of that substance having a total mass of 7.5 g. What is
needed, clearly, is a very large number by which elementary particles
must be multiplied in order to yield a mass whose value in grams is
equal to the value, in amu, of its average atomic mass. This is known
as Avogadro's number.

AVOGADRO'S NUMBER.

The first scientist to recognize a meaningful distinction between
atoms and molecules was Italian physicist Amedeo Avogadro (1776-1856).
Avogadro maintained that gases consisted of particles—which he
called molecules—that in turn consisted of one or more smaller
particles. He further reasoned that one liter of any gas must contain
the same number of particles as a liter of another gas.

In order to discuss the behavior of molecules, it was necessary to set
a large quantity as a basic unit, since molecules themselves are very
small. This led to the establishment of what is known as
Avogadro's number, equal to 6.022137 × 10
23
(more than 600 billion trillion.)

The magnitude of Avogadro's number is almost inconceivable. The
same number of grains of sand would cover the entire surface of Earth
at a depth of several feet. The same number of seconds, for instance,
is about 800,000 times as long as the age of the universe (20 billion
years). Avogadro's number—named after the man who
introduced the concept of the molecule, but only calculated years
after his death—serves a very useful purpose in computations
involving molecules.

THE MOLE.

To compare two substances containing the same number of atoms or
molecules, scientists use the mole, the SI fundamental unit for
"amount of substance." A mole (abbreviated mol) is,
generally speaking, Avogadro's number of atoms or molecules;
however, in the more precise SI definition, a mole is equal to the
number of carbon atoms in 12.01 g (0.03 lb) of carbon. Note that, as
stated earlier, carbon has an average atomic mass of 12.01 amu. This
is no coincidence, of course: multiplication of the average atomic
mass by Avogadro's number yields a figure in grams equal to the
value of the average atomic mass in amu.

The term "mole" can be used in the same way we use the
word "dozen." Just as "a dozen" can refer
to twelve cakes or twelve chickens, so "mole" always
describes the same number of molecules. Just as one liter of water, or
one liter of mercury, has a certain mass, a mole of any given
substance has its own particular mass, expressed in grams. A mole of
helium, for instance, has a mass of 4.003 g (0.01 lb), whereas a mole
of iron is 55.85 g (0.12 lb) These figures represent the molar mass
for each: that is, the mass of 1 mol of a given substance.

Once again, the value of molar mass in grams is the same as that of
the average atomic mass in amu. Also, it should be clear that, given
the fact that helium weighs much less than air—the reason why
helium-filled balloons float—a quantity of helium with a mass
of 4.003 g must be a great deal of helium. And indeed, as indicated
earlier, the quantity of atoms or molecules in a mole is sufficiently
great to make a sample that is large, but still usable for the
purposes of study or comparison.

Measuring Volume

Mass, because of its fundamental nature, is sometimes hard to
comprehend, and density requires an explanation in terms of mass and
volume. Volume, on the other hand, appears to be quite
straightforward—and it is, when one is describing a solid of
regular shape. In other situations, however, volume measurement is
more complicated.

As noted earlier, the volume of a cube can be obtained simply by
multiplying length by width by height. There are other means for
measuring the volume of other straight-sided objects, such as a
pyramid. Still other formulae, which make use of the constant π
(roughly equal to 3.14) are necessary for measuring the volume of a
cylinder, a sphere, or a cone.

For an object that is irregular in shape, however, one may have to
employ calculus—but the most basic method is simply to immerse
the object in water. This procedure involves measuring the volume of
the water before and after immersion, and calculating the difference.
Of course, the object being measured cannot be water-soluble; if it
is, its volume must be measured in a non-water-based liquid such as
alcohol.

LIQUID AND GAS VOLUME.

Measuring liquid volumes is even easier than for solids, given the
fact that liquids have no definite shape, and will simply take the
shape of the container in which they are placed. Gases are similar to
liquids in the sense that they expand to fit their container; however,
measurement of gas volume is a more involved process than that used to
measure either liquids or solids, because gases are highly responsive
to changes in temperature and pressure.

If the temperature of water is raised from its freezing point to its
boiling point—from 32°F (0°C) to 212°F
(100°C)—its volume will increase by only 2%. If its
pressure is doubled from 1 atm (defined as normal air pressure at sea
level) to 2 atm, volume will decrease by only 0.01%. Yet if air were
heated from 32° to 212°F, its volume would increase by 37%;
if its pressure were doubled from 1 atm to 2, its volume would
decrease by 50%.

Not only do gases respond dramatically to changes in temperature and
pressure, but also, gas molecules tend to be non-attractive toward one
another—that is, they tend not to stick together. Hence, the
concept of "volume" in relation to a gas is essentially
meaningless unless its temperature and pressure are known.

Comparing Densities

In the discussion of molar mass above, helium and iron were compared,
and we saw that the mass of a mole of iron was about 14 times as great
as that of a mole of helium. This may seem like a fairly small factor
of difference between them: after all, helium floats on air, whereas
iron (unless it is arranged in just the right way, for instance, in a
tanker) sinks to the bottom of the ocean. But be careful: the
comparison of molar mass is only an expression of the mass of a helium
atom as compared to the mass of an iron atom. It makes no reference to
density, which is the ratio of mass to volume.

Expressed in terms of the ratio of mass to volume, the difference
between helium and iron becomes much more pronounced. Suppose, on the
one hand, one had a gallon jug filled with iron. How many gallons of
helium does it take to equal the mass of the iron? Fourteen? Try
again: it takes more than 43,000 gallons of helium to equal the mass
of the iron in one gallon jug! Clearly, what this shows is that the
density of iron is much, much greater than that of helium.

This, of course, is hardly a surprising revelation; still, it is
sometimes easy to get confused by comparisons of mass as opposed to
comparisons of density. One might even get tricked by the old
elementary-school brain-teaser that goes something like this:
"Which is heavier, a ton of feathers or a ton of
cannonballs?" Of course neither is heavier, but the trick
element in the question relates to the fact that it takes a much
greater volume of feathers (measured in cubic feet, for instance) than
of cannonballs to equal a ton.

One of the interesting things about density, as distinguished from
mass and volume, is that it has nothing to do with the amount of
material. A kilogram of iron differs from 10 kg of iron both in mass
and volume, but the density of both samples is the same. Indeed, as
discussed below, the known densities of various materials make it
possible to determine whether a sample of that material is genuine.

COMPARING DENSITIES.

As noted several times, the densities of numerous materials are known
quantities, and can be easily compared. Some examples of density, all
expressed in terms of grams per cubic centimeter, are listed below.
These figures are measured at a temperature of 68°F (20°C),
and for hydrogen and oxygen, the value was obtained at normal
atmospheric pressure (1 atm):

Comparisons of Densities for Various Substances:

Oxygen: 0.00133 g/cm
3

Hydrogen: 0.000084 g/cm
3

Ethyl alcohol: 0.79 g/cm
3

Ice: 0.920 g/cm
3

Water: 1.00 g/cm
3

Concrete: 2.3 g/cm
3

Iron: 7.87 g/cm
3

Lead: 11.34 g/cm
3

Gold: 19.32 g/cm
3

Specific Gravity

IS IT REALLY GOLD?

Note that pure water (as opposed to sea water, which is 3% more dense)
has a density of 1.0 g per cubic centimeter. Water is thus a useful
standard for measuring the specific gravity of other substances, or
the ratio between the density of that substance and the density of
water. Since the specific gravity of water is 1.00—also the
density of water in g/cm
3
—the specific gravity of any substance (a number, rather than a
number combined with a unit of measure) is the same as the value of
its own density in g/cm
3
.

Comparison of densities make it possible to determine whether a piece
of jewelry alleged to be solid gold is really genuine. To determine
the answer, one must drop the sample in a beaker of water with
graduated units of measure clearly marked. Suppose the item has a mass
of 10 g. The density of gold is 19 g/cm
3
, and because density is equal to mass divided by volume, the volume
of water displaced should be equal to the mass divided by the density.
The latter figure is equal to 10 g divided by 19 g/cm
3
, or 0.53 ml. Suppose that instead, the item displaced 0.88 ml of
water. Clearly it is not gold, but what is it?

Given the figures for mass and volume, its density is equal to 11.34
g/cm
3
—which happens to be the density of lead. If, on the other
hand, the amount of water displaced were somewhere between the values
for pure gold and pure lead, one could calculate what portion of the
item was gold and which lead. It is possible, of course, that it could
contain some other metal, but given the high specific gravity of lead,
and the fact that its density is relatively close to that of gold,
lead is a favorite gold substitute among jewelry counterfeiters.

SPECIFIC GRAVITY AND THE DENSITIES OF PLANETS.

Most rocks near the surface of Earth have a specific gravity somewhere
between 2 and 3, while the specific gravity of the planet itself is
about 5. How do scientists know that the density of Earth is around 5
g/cm
3
? The computation is fairly simple, given the fact that the mass and
volume of the planet are known. And given the fact that most of what
lies close to Earth's surface—sea water, soil,
rocks—has a specific gravity well below 5, it is clear that
Earth's interior must contain high-density materials, such as
nickel or iron. In the same way, calculations regarding the density of
other objects in the Solar System provide a clue as to their interior
composition.

This brings the discussion back around to a topic raised much earlier
in this essay, when comparing the weight of a person on Earth versus
that person's weight on the Moon. It so happens that the Moon
is smaller than Earth, but that is not the reason it exerts less
gravitational pull: as noted earlier, the gravitational force a
planet, moon, or other body exerts is related to its mass, not its
size.

It so happens, too, that Jupiter is much larger than Earth, and that
it exerts a gravitational pull much greater than that of Earth. This
is because it has a mass many times as great as Earth's. But
what about Saturn, the second-largest planet in the Solar System? In
size it is only about 17% smaller than Jupiter, and both are much,
much larger than Earth. Yet a person would weigh much less on Saturn
than on Jupiter, because Saturn has a mass much smaller than
Jupiter's. Given the close relation in size between the two
planets, it is clear that Saturn has a much lower density than
Jupiter, or in fact even Earth: the great ringed planet has a specific
gravity of less than 1.